Question: There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle?
Solution: We can list the first 10 rows of Pascal's triangle, and mark the even numbers.

[asy]
usepackage("amsmath");
unitsize(0.5 cm);

int i, j, n;

for (int i = 0; i <= 9; ++i) {
for (int j = 0; j <= 9; ++j) {
  if (i + j <= 9) {
    n = choose(i + j,i);
    if (n % 2 == 0) {label("$\boxed{" + string(n) + "}$", i*(-1,-1) + j*(1,-1));}
    if (n % 2 == 1) {label("$" + string(n) + "$", i*(-1,-1) + j*(1,-1));}
  }
}}
[/asy]

Thus, the number of even numbers is $1 + 3 + 2 + 3 + 7 + 6 = \boxed{22}.$